DESIGN OF REINFORCED CONCRETE SHELL
STRUCTURES AND FOLDED PLATES
( First Revision)
First Reprint JANUARY 1992
-UDC 624’012’45 - 04 : 624’07’4/‘42
BUREAU OF INDIAN STANDARDS
MANAK BHAVAN, 9 BAHADUR SHAH ZAFAR MARG
NEW DELHI 110 002
Gr 7 November 1989
( Reaffirmed 1998 )
IS :2210 - 1988
DESIGN OF REINFORCED CONCRETE SHELL
STRUCTURES AND FOLDED PLATES
(First Revision )
0.1 This Indian Standard ( First Revision) was
adopted by the Bureau of Indian Standards on I5
November 1988, after the draft finalized by the
Criteria for Design of Special Structures Sectional
Committee had been approved by the Civil Engineer-
ing Division Council.
0.2 Shells and folded plates belong to the class of
stressed-skin structures which, because of their
geometry and small flexural rigidity of the skin,
tend to carry loads primarily by direct stresses acting
in their plane. Wherever shell is referred to in this
standar$ it refers to thin shell. On account of
multiphclty of the types of reinforced concrete shell
and folded plate structures used in present day
building practice for a variety of applications demand-
ing roofing of large column-free area, it is not
practicable to lay down a rigid code of practice to
cover all situations. Therefore, this standard lays
down certain general recommendations for the
guidance of the designers. .*
0.3 Cylindrical shells have been in use in building
construction for the past six decades. Well developed
theories exist for their analysis. Labour-saving
short cuts, such as, charts and tables for their
design, are also available.
0.4 Although shells of double curvature, with the
exception of domes, have been introduced on a large
scale comparatively recently into building construc-
tion, these are likely to be used more and more in
future. Being non-developable surfaces, they are
more resistant to buckling than cylindrical sheTls
and in general, require less thickness. This saving
in materials is, however, often offset by the datively
expensive shuttering required for casting them.
Among the doubly-curved shells, ’ the hyperbolic
paraboloid and the conoid have, however, the
advantage of less expensive shuttering because their
ruled surfaces can be formed by straight plank
0.5 Folded plates are often competitive with shells
for covering large column-free areas. They usually
consume relatively more materials compared to
shell but this disadvantage is often offset by the
simpler framework required for their construction.
0.6 This standard was first published. in 1962. The
present revision is based on the developments in the
design of shell and folded plate structures subsequent-
ly and to include more rigorous methods of analysis
which have become available to enforce more
rational criteria of design.
0.7 For the purpose of deciding whether a particular
requirement of this standard is complied with, the
final value, observed or calculated, expressing the
result of a test or analysis, shall be rounded off in
accordance with IS : 2-1960*. The number of
significant places retained in the rounded off value
should be the same as that of the specified value in
*Rules for rounding off numerical values (revised ).
1.1 This standard lays down recommendations for
the classification, dimensional proportioning, analysis
and design of cast in situ, reinforced concrete thin
shells and folded plates. This standard does not
deal with construction practices relating to these
structures which have been separately dealt in
IS : 2204-1962*.
2.0 For the purpose of this standard, the following
definitions shall apply.
2.1 Asymmetrical Cylindrical Shells - Cylindrical
shells which are asymmetrical about the crown.
*Code of practice for construction of reinforced concrete
2.2 Barrel Shells - Cylindrical shells which are
symmetrical about the crown (see Fig. 1).
2.3 Butterfly Shells - Butterfly shells are those
which consist of two parts of a cylindrical shell
joined together at their lower edges (see Fig. 2).
2.4 Chord Width - The chord width is the horizon-
tal projection of the arc of the cylindrical shell.
2.5 Continuous Cylindrical Shells - Cylindrical shells
which are longitudinally continuous over the
traverses (see Fig. 3 ).
NOTE- Doubly-curved shells continuous in one or both
directions may be termed as continuous shells.
2.6 Cylindrical Shells - Shells in which either the
directrix or generatrix is a straight line.
2.7 Edge Member - A member provided at the
edge of a shell.
NOTE- Edge members increase the rigidity of the shell
edge and h&p in accommodating the reinforcement.
2.8 End Frames or Traverses - End frames or
traverses are structures provided to support and
preserve the geometry of the shell.
NOTE- They may be solid diaphragms, arch ribs, portal
frames or bowstring girders. Where a clear soffit is
required, specially for the use of movable formwork, the
end frames may consist of upstand ribs.
2.9 Folded Plates - Folded plates consist of a series
of thin plates, usually rectangul c, joined monolithi-
cally along their common e2ges and supported on
diaphragms. They are also known as hipped plates.
NOTE- Shapes of Folded Plates and Their Applications-
A few of the commonly used shapes of folded plates are
shown in Fig. 4. The simplest is V-shaped unit (Fig.
4A) but this may not provide enough area of concrete at
the top and bottom to resist the compressive forces due
to bending and to accommodate the reinforcement. The
trough-shaPed or the trapezoidal unit (Fig. 4B and 4E)
eliminates _ these disadvantages. Asy&meirical section;
of the ‘2’ shape (Fig. 4C) provided with window glazing
between two adjacent units serve as north-light roofs for
factory buildings. The shape shown in Fig. 4D is
obtained by replacing the curved cross-se&x of a
cylindrical shell by a series of straight plates. This has
the advantage of greater structural depth compared to
other shapes. Buttetiy type of folded plates shown in
Fig. 4F are also employed to cover factory roofs as there
are provisions for window glazing. Tapering plates are
also used as roofs mainly for aesthetic reasons. I-lipped
plate structures of the pyramidal types are used for tent-
shaped roofs, cooling towers, etc.
2.10 Gauss Curvature - The product of the two
principal curvatures, ~/RIand l/R1 at any point on
the surface of the shell.
2.11 Generatrix, Directrix - A curve which moves
parallel to itself over a stationary curve generates a
surface. The moving curve is called the generatrix
and the stationary curve the directrix. One of them
may be a straight line.
NOTE- The common curves used for cylindrical shells
are, arc of a circle, semi-ellipse, parabola, catenary and
2.12 Junction Member - The common edge mem-
ber at the junction of two adjacent shells.
2.13 Multiple Cylindrical Shells - A series of
parallel cylindrical shells w.lich are transversely
2.14 North-Light Shells - Cylindrical shells with two
springings at different levels and having provisions
for north-light glazing (see Fig. 5).
2.15 Radius - Radius at any point of the. shell- in
one of the two principal directions.
NOTE- If cylindrical shell of a circular arc is used, the
radius of the arc is the radius of the shell. In other cases,
the radius R at any point is related to the radius &, at thd
crown by R = R, cos n#, where 4 is the angle of inclination
IS : 2210- 1988
of the tangent to the curve at that point. The parabola,
value ofn is 1, -2 and -3 for cycloid, catenary and para-
bola, respectively. For an ellipse:
(a’ sin*+ + b* c&d )*I*
where a and b are the semi-major and semi-minor axes,
respectively, and d is the slope of the tangent at the point.
2.16 Rise - The vertical distance between the apex
of the curve representing the centre line of the shell
and the lower most springing.
2.17 Ruled Surfaces - Surfaces which can be
generated entirely by straight lines. The surface is
said to be ‘singly ruled’ if at every point, a single
straight line only can be ruled and ‘doubly ruled’ if
at every point, two straiglt lines can be ruled.
Cylindrical shells, conical shells and conoids are
examples of singly ruled surfaces; hyperbolic para-
bol jids and hyperboloids of revolution of one sheet
are examples of doubly ruled surface ( see Fig. 6 ).
2.18 Semi-Central Angle - Half the angle subtend-
ed by the arc of a symmetrical circular shell at the
2.19 Shells - Thin shells are those in which the
radius to thickness ratio should not be more than 20.
2.20 Shells of Revolation - Shells which are obtain-
ed when a plane curve is rotated about the axis of
symmetry. Examples are segmental domes, cones,
parab Jloids of revolution, hyperboloids OI revolu-
tion, etc ( see Fig. 7 ).
2.21 Shells of Translation - Shells which are
obtained when the plane of the generatrix and the
directrix are at right angles. Examples are cylindri-
cal shells, elliptic paraboloids, hyperbolic paraboloids,
etc ( see Fig. 8 ) ,
2.22 span - The span of a cylindrical shell is the
distance between the centre lines of two adjacent
end frames of traverses (see Fig. 1).
3.1 For the purpose of this standard, unless other-
wise defined in the text, the following notations
shall have the meaning indicated against each:
semi-major axis of an elliptical shell;
semi-minor axis of an elliptical shell;
thickness of shell;
Modulus of elasticity of concrete (long
Modulus of elasticity of steel;
stress function which gives the in-plane
stress in doubly-curved shells when
bending is also considered;
characteristic strength of concrete;
critical buckling stress;
rise of shell;
f cc =
permissible compressive stress from
total depth of shell, measured from the
crown of the shell to the bottom of the
bending moment in the shell in the
My = bending moment in the shell in the
twisting moment in the shell;
= real membrane stresses in the shell;
= projected membrane forces;
permissible buckling load per unit area
of the surface of doubly-curved shells;
n =. --
r = 62;
R = radius;
& = radius at crown;
R, and Rx = principal radii of curvature at any
point on the surface of shell;
s = shear stress;
Tx = normal stress in the x-direction;
TV = normal stress in the y-direction;
Wx, WV and Wx = real forces on unit area of
the shell in the x, y and z-
IV = de&&on in the direction of z-axis;
X, YandZ = fictitious forces on unit projected
area of the shell in the X, y and z-
X, y and z = axes of co-ordinates;
Q, = stress function used in the membrane
analysis of doubly-curved shell;
9 = angle of inclination of tangent to the
curve at any point;
+c = semi-central angle of a symmetrical
circular cylindrical shell;
P and K = Aas Jakobsen’s varameters for
cylindrical shells; _
Poisson’s ratio; and
4. CLASSIFICATION OF SHELLS
4.1 General - Shells may be broadly classified as
‘singly-curved’ and ‘doubly-curved’. This is based
on Gauss curvature. The gauss curvature of singly-
curved shells is zero because one of their principal
curvatures is zero. They are, therefore, developable.
Doubly-curved shells are non-developable and are
classified as synclastic or anticlastic according as
their Gauss curvature is positive or negative.
4.1.1 The governing equations of membrane
theory of singly curved shells are parabolic. It is
-elliptic for synclastic shells and hyperbolic for
anticlastic shells. If z = f (x, y) is the equation
to the surface of a shell, the surface will be synclas-
tic, developable or anticlastic according as s2-rf 4
0 where t, s and t are as defined in 3.1.
4.1.2 There are other special types of doubly
curved shells, such as, funicular shells, which are
synclastic and anticlastic in parts and corrugated
shells which are alternately synclastic and anticlastic.
The gauss curvature for such shells is positive
where they are synclastic and negative where they
4.2 The detailed classification of shell structures is
given in Appendix A.
5.1 Con&&e - Controlled concrete shall be used
for all shell. and folded plate structures. The
concrete is of minimum grade M20. The quality
of materials used in concrete, the methods of pro-
portioning and mixing the concrete shall be done
in accordance with the relevant provisions of
IS : 456-1978*.
NOTE-High cement content mixes are generally
undesirable as they shrink excessively giving rise to
5.2 Steel - The steel for the reinforcement shall be:
a) mild steel and medium tensile steel bars and
hard-drawn steel wire conforming to IS : 432
(Part I)-1982 and IS : 432 (Part 2)-1982t;
b) hard-drawn steel. wire fabric for concrete
Ti$forcement conforming to IS : 1566-19821;
c) high strength deformed bars conforming to
IS : 1786-1985s.
5.2.1 Welding may be used in .reinforccment in
accordance with IS : 456-1978*.
*Code of practice for plain and reinforced concrete
( third revision ).
tSpecification for mild steel and medium tensile steel bars
and hard-drawn steel wire for concrete reinforcement:
Part 1 Mild steel and medium tensile steel bars (third
rcvisfon ) .
Part 2 Hard-drawn steel wire (third revision).
SSpecification for hard-drawn steel wire fabric for
concrete reinforcement (second revision ).
#Specification for high strength deformed steel bars and
wires for concrete reinforcement ( rhfrdrevlsfon).
Non+ Diaphragms 4F
e‘IG- 4 FOLDED PLAYnoiSbOWO,
68 HYPERBOLIC PARABOLOID
6C HYPERBOLOID OF REVOLUTION OF ONE SHEET
FIG. 6 RULED SURFACES
Is : 2210- 1988
7A SEGMENTAL DOME
78 PARABOLOID OF REVOLUTION
Fro. 7 !fhLU OFREVOLUTION
8A ELLIPTIC PARABOLOID
88 HYPERBOLIC PARABOLOID
FIG. 8 SHELLS OF TRANSLATION
IS : 2210 - 1988
6.1 Unless otherwise specified, shells and folded
plates shall be designed to resist the following load
a) Dead load,
b) Dead load + appropriate live load or snow
c) Dead load + appropriate live load -I- wind
d) Dead load + appropriate live load + seismic
6.2 Dead loads shall be calculated on the basis of
the unit weights taken in accordance with IS : 875
6.3 Live loads, wind loads and snow loads shall be
taken as specified in IS : 875 (Parts 2 to 4)-1987*.
6.4 Seismic loads shall be taken in accordance with
IS : 1893-1984t.
6.5 Where concentrated loads occur, special consi-
derations should be given in analysis and design.
7. SELECTION OF DIMENSIONS
7.1.1 Thickness of Shells - Thickness of shells
shall not normally be less than 50 mm if singly-
curved and 40 mm if doubly-curved. This require-
ment does not, however, apply to small precast con-
crete shell units in which the thickness may be less
than that specified above but it shall in no case be
less than 25 mm (see IS : 6332-19842).
22.214.171.124 The reinforcement shall have a minimum
clear cover of 15 mm or its nominal size whichever
7.1.2 Shells are usually thickened for some dis-
tance from their junction with edge members and
traverses. The thickening is usually of the order
of 30 percent of the shell thickness. It is, however,
important to note that undue thickening is undesir-
able. In the case ‘of singly-curved shells, the dis-
tance over which the thickening at the junction of
the shell and traverse is made should be between
0’38 4R.d and 0’76 1/rd, where R and d are the
radius and the thickness, respectively. The thicken-
ing of shell at straight edges shall depend on the
*Code of practice for design loads (other than earth-
quake) for building and structures:
Part 1 Dead loads (second revision ).
Part 2 Imposed loads (second revision ).
Part 3 Wind loads (second revision ).
Part 4 Snow loads (secotlri revision ).
tCriteria for earthquake resistant design of structures
(fourth revision ) .
*Code of practice for construction of floors and roofs
using Precast doubly-curved shell units (first revislon ).
transverse bending moment. For doubly-curved
shells, this distance will depend upon the geometry
of the shell and the boundary conditions as the ex-
tent of bending penetration is governed by these
7.1.3 Thickness of Folded Plates - The thickness
of folded plates shall not normally be less than
7.2 Other Dimensions
7.2.1 CyIindrical Shells
126.96.36.199 The span should preferably be less
than 30 m. Shells longer than 30 m will involve
special design considerations, such as the application
of prestressing techniques.
188.8.131.52 The width of the edge member shall
generally be limited to’ three times the thickness of
184.108.40.206 The radius of shell structures shall be
selected keeping acoustic requirements in view.
Coincidence of the centre of curvature with the
working level should be avoided unless suitable
acoustic correction is made. It is, however, impor-
tant to note that even where coincidence of centre of
curvature with the working level is avoided, acoustic
treatment may be necessary iu certain cases.
220.127.116.11 A single cylindrical shell whose span is
larger than three times the chord width shall have a
total depth, H, between l/6 and l/l2 of its span
(the former value being applicable to smaller spans).
The rise in the case of a shell without edge members
shall not be less than l/l0 of its span.
18.104.22.168 For a shell with chord width larger than
three times the span, the rise of the shell shall not
be less than l/8 of its chord width.
22.214.171.124 The chord width of shells shall pre-
ferably be restricted to six times the span as other-
wise arch action is likely to predominate.
126.96.36.199 The semi-central angle shall preferably
be between 30 and 40”.
NOTE - Keeping the semi-central angle between these
limits is advisable for the following reasons:
a) If the angle is below 40°, the effect of wind load
on the shell produces only suction; and
b) With slopes steeper than 40”. backforms may
Within these limits the semi-central angle shall
be as high as possible consistent with the functional
7.2.2 Folded Plates - For folded plates of type
shown in Fig. 4D, the selection of depth may be
based on the rules applicable to cylindrical shells.
With other shapes, such as, the ‘V’ or the trough,
the depth may be taken as about l/l5 of the span
for preliminary designs.
188.8.131.52 The angle of inclination of the plates to
horizontal shall be limited to about 40 for in-situ
construction in order to facilitate placing of concrete
without the use of the backforms.
8.0 General - Shells may be analyzed either by
linear elastic analysis based on theory of elasticity or
yield line theory. Methods based on yield line
theory for shells are still the subject of research and
experimentation and, therefore, for the present, it
is recommended that they may be used along with
model tests to check the load carrying capacity.
The finite element method has become a practi-
cal and popular method of analysis for all types of
structures. Many common and important features
of shell and folded plate structures that cannot be
considered by classical methods can now be analyzed
satisfactorily by the finite element method. For
Complex support or boundary conditions;
Openings large enough to disturb global
Irregular surface geometry;
Highly variable or localized loads;
Tapering folded plates or boxes or silo and
Heavy and eccentric stiffners;
Irregular surface geometry;
or any inelastic
Sudden changes in curvature;
Shells under dynamic wind action; and
Possible effect of settlement.
When one or more of these complexities occur in
shell structures, it is advisable to use finite element
method, at least for a final acceptance of the design.
Even normal shell structures of spans larger than
30 m should be analyzed by finite element method
if it is expected that there would be serious and
significant structural participation in the shell be-
haviour by the supporting units, such as, edge and
intermediate beams, stiffners, and or intermediate
traverses ( specially flexible traverses ) cable supports,
For many shells, finite strip method ( FSM) ( a
particular form of finite element method ) of analysis
is easier to apply and also more economical to use
than the finite element method. All types of
prismatic folded plates and cylindrical shells can be
IS : 2210 - 1988
handled by FSM since these shells can be discretized
into long strip elements. Shells of revolution can
also be efficiently analyzed by this method, after
discretization of such shells into finite ring elements.
Since FSM can be used even when the loads are not
uniform, it may be advisable to use the method even
for simple shells that are amenable to analysis by
common classical methods.
The common classical methods of analysis of
shells are mentioned in the following clauses for
use for the analysis of common types of shells that
are without any of the complexities.
8.1 Cylindrical Shells
8.1.1 Analytical Methodr - The analytical methods
consist of two parts, membrane analysis and edge
184.108.40.206 Membrane analysis - In the membrane
analysis, the shell is regarded as a perfectly flexible
membrane which is infinite in extent and is assumed
to carry loads by means of forces in its plane only.
This analysis gives the two normal stress resultants
Nx and NY in the longitudinal and the transverse
directions and the shear stress resultant Nxv.
220.127.116.11 Edge disturbance analysis - Shells, in
practice, are always limited by finite boundaries
where the boundary conditions demanded by the
membrane theory are not obtained with the result
that a pure membrane state would seldom exist.
Edge disturbances emanate from the boundaries,
altering the membrane state and causing bending
stresses in the shell. These are accounted for bye,
carrying out the edge disturbance analysis. Usually
edge disturbance analysis is confined to disturbance
emanating from straight edge as any disturbance
emanating from curved edges is damped quite fast.
Even in the case of disturbance from straight edges,
the bending stresses would get damped out more
rapidly in shells having chord width larger than the
span and would seldom travel beyond the crown,
with the result that the effect of the further edge
may be ignored without appreciable error.
The superposition of the membrane and the edge
disturbance stresses gives the final stress pattern in
18.104.22.168 Tables for the analysis of circular cylin-
drical shells - Simplifications in the analysis -of
shells are possible by systematizing the calculations 4
making use of tables compiled for this purpose
(see Appendix B ).
8.1.2 Applicability of the Methods of Analysis
22.214.171.124 Cylindrical shells with k ratio less than
rr shall be analyzed using any of the accepted
analytical methods (see Appendix B ).
IS : 2210- 1988
In such shells, if P exceeds 10 and K exceeds 0’15,
the effect at any point on the shell of the dis-
turbances emanating from the farther edge may be
12 -’ R" w’ R’
and K = -
For shells with P less than 7 and K less than 0’12,
the effect of the disturbances from both the edges
shall be considered. Shells with P values between 7
and 10 and K between 0’12 and 0’15 are relatively
infrequent. However, should such cases arise, the
effects of both the edges shall be considered.
126.96.36.199 Cylindrical shells with L/R greater than
or equal to n may be treated as beams of curved
cross section spanning between the traverses and the
analysis carried out using an approximate method
known as the beam method (see Appendix B)
which consists of the following two parts:
a) The beam calculation which gives the
longitudinal stress resultant N, and the shear
stress resultant NXY,and
b) The arch calculation which gives the transverse
stress resultant NY and the transverse moment
8.1.3 Continuous Cylindrical Shells
188.8.131.52 Analytical methods - In the analytical
methods, the problem of continuous shells is solved
in two stages. In the first, the shell is assumed
to be ‘simply supported over one span and
all the stress resultants worked out. In the second
stage, correcg;ns for continuity are worked out and
are superimposed on the values corresponding to the
simply supported span. Long cylindrical shells can
be analyzed by approximating the cylindrical profile
by a folded plate shape and applying well known
analysis methods for continuous folded plates (see
184.108.40.206 Beam method - Solution of continuous
shell with -$ 2 ?r is simpler by the beam method.
The bending moment factors are obtained by solving
the corresponding continuous beam. Thereafter,
analysis can be continued as in 220.127.116.11.
8.2 Donhly-Curved SheIIs
8.2.1 Membrane Analysis - In the membrane
analysis, it is assumed that the shell carry loads by
in-plane stress resultants and usually only deep
doubly-curved shells behave like membranes. The
governing equations for membrane analysis of doubly-
curved are usually solved by using stress functions
( see Appendix C ).
8.2.2 Shallow Shells - Shells may be considered
shallow if the rise to span ratio is less than or equal
to l/5 and p2 and q’ may be ignored in all the ex-
pressions. The shorter side shall be considered as
the span for this purpose for shells of rectangular
g*ound plan. A shell with a circular ground plan
may beconsidered shallow if the rise does not exceed
l/5 of the diameter.
NATE- Based on the same consideration of ignoring
p’. q’ and P q being very small, it is sometimes suggested
that the shells may be treated as shallow if the surface
is such that the values ofp and q do not exceed l/8 at
any point on it. However, in normal cases, for practical
purposes, higher values ofp and q up to half may be
considered as shallow provided the span to rise ratio does
not exceed 5.
8.2.3 Boundary Conditions for Doubly-Curved
Shells - In general, in the membrane analysis of
synclastic shells, only one boundary condition is
admissible on each boundary. For an anticlastic
shell, the boundary conditions have to be specified
in a special manner as the characteristic lines of
such surfaces play a significant role in the membrane
theory. The type of boundary conditions that can
be specified depend on whether or not the boundaries
of the shell are characteristic lines.
Further, a membrane state of stress can be
maintained in a shell only if the boundaries are such
that the reactions exerted by the boundary members
on the shells correspond to stresses in the shell at
the boundaries given by the membrane theory. It
is seldom possible to provide boundary conditions
which would lead to a pure membrane state of stress
in the shell. In most practical cases, a resort to
bending theory becomes necessary.
18.104.22.168 Only deep doubly-curved shells behave
like membranes and it is only for such shells that
a membrane analysis is generally adequate for design
(see also 8.0). Bending analysis is necessary for
all singly-curved shells and shallow doubly-curved
shells. The governing equation for bending analysis
of shallow shells are given in AppendixC.
8.2.4 Bending Theory of Doubly-Curved Shells -
The governing equations for bending analysis of
shallow doubly-curved shells are
8.2.5 Bending ?heory of Shells of Revolution,
Symmetrically Loaded - In synclastic shells of
revolution, such as, domes which are symmetrically
loaded, a more or less membrane state of stress
exists. The bending stresses are confined to a very
narrow strip close to edge members and get rapidly
damped out. The bending stresses in domes can be
calculated with sufficient accuracy by approximate
methods like Geckler’s approximation.
0.2.6 Funicular Shells - The shapes of these shells
are so choosen that, under uniformly distributed
vertical loads, in a membrane state of stress, they
develop only pure compression unaccompanied by
shear stresses. Thus theoretically no reinforcement
will be necessary except in the edge members. Small
precast funicular shells without any reinforcement
except in edge beams are suitable for roofs and
floors of residential, industrial, and institutional
buildings (see IS : 6332’1984* ). For roofs of larger
size, in situ construction may be resorted to; in such
shells, provision of reinforcement is necessary to take
care of the effects of shrinkage, temperature and
*Code of practice for construction of floors and roofs
using precast doubly-curved shell units (fist revision ).
8.4 Folded Plates - The structural action of folded
plates may be thought of as consisting of two parts.
the ‘slab action’ and the ‘plate action’. By the slab
action, the loads are transmitted to the joints by the
transverse bending of the slabs. The slabs, because of
their large depth and relatively small thickness, offer
considerable resistance to bending in their own planes
and are flexible out of their planes. The loads are,
therefore, carried to the end diaphragms by the
longitudinal bending of the slabs in their own planes.
This is known as ‘plate action’. The analysis of
folded slabs is carried out in two stages.
83.1 Transverse Slabs Action Analysis - The
transverse section of the slab, of unit length, is
analyzed as a continuous beam on rigid supports.
The joint loads obtained from this analysis are
replaced by their components in the planes of the
slabs and these are known as plate loads.
8.3.2 In Plane Plate Action Analysis - Under the
action of ‘plate loads’ obtained above, each slab is
assumed to bend independently between the dia-
phragms, and the longitudinal stresses at the edges
are calculated. Continuity demands that the
longitudinal stresses at the common edges of the
adjacent slabs be equal. The corrected stresses are
obtained by introducing edge shear forces.
8.4 Expansion Joints - The expansion joint shall
conform to provisions laid down in IS : 456-1978*.
In the case of folded plates, it is recommended that
the joint may be located in the ridge slab. In the
cases of large spans where it is not feasible to pro-
vide expansion joints, effects of shrinkage shall be
taken care of in the design.
8.5 Openings in Shells - Openings in shells shall
preferably be avoided in zones of critical stresses.
Small openings of size not exceeding five times the
thickness in shells may be treated in the same way
as in the case of reinforced concrete structures. For
larger openings, detailed analysis should be carried
out to arrive at stresses due to the openings.
9. ELASTIC STABILITY
9.1 Permissible Stresses
9.1.1 Permissible stresses in steel reinforcement,
and concrete for shells and folded plates shall
be in accordance with the provisions given in
1s : 456-1978*.
9.2 Causes of Instability - Instability in a cylindri-
cal shell may be caused by:
a) local buckling in zones submitted to com-
b) flattening of shells, known as the ‘Brazier
Effect’, which occurs particularly in shells
without edge members; and
*Code of practice for plain and reinforced concrete
(third revision ).
IS : 2210 - 1988
instability caused by the combined effect of
bending and torsion in the shell as a whole.
This occurs particularly in asymmetrical
9.3 Buckling in Cylindrical Shells
9.3.1 The permissible buckling stress fatin cylind-
rical shells shall be calculated as follows:
f ac =
characteristic strength of concrete at 28
critical buckling stress determined in
accordance with ( a ), ( b ) and ( c ) below:
Shells with P < 7 and K < O’I2 - In such shells,
buckling is caused by excessive longitudinal
compression near the crown of the shell and the
critical buckling stress fcrshall be calculated as
fcr = 0.20%
Shells with P > IO and K > 0’15 - In such
shells, the transverse stresses tend to be critical
from the point of view of buckling and the
critical buckling stress fcrshall be determined as
1) For shells with L < 2’3qdF
fcr = EC [ 3’4 (+ )’ + 0’025 ( +) ]
2) For shells with L > 2’31/dy
l- 1’18 d$
EC = modulus of elasticity of concrete, which
may be taken as = 2 x 3fck;
E, = modulus of elasticity of steel;
d = thickness of the shell; and
R = radius of curvature.
Shells with P values between 7 and 10 and
K between 0’12 and 0’15 are relatively infre-
quent. For such shells, formulae given in
(a) or (b) shall apply depending upon whether
longitudinal or transverse stresses are critical
from Considerations of elastic stability.
The value of modulus of elasticity of concrete
to be used in the above formulae for calculating the
buckling stresses should be the value for long term
modulus including the effect of creep also.
IS : 2210 - 1988
9.4 Buckling in Doubly-Carved Shells - For
spherical shells, the permissible buckling load per
unit area of surface, P,,,,, from considerations of
elastic stability, is given by:
ECd and R are as defined in 3.1.
For other types of doubly-curved shells, the
permissible buckling load per unit area of surface,
Ppermshall be calculated from the formula:
R, and R, are principal radii of curvature at
any point, and EC and d are as defined
9.5 Bwkhg in Folded Plates - The folded plate
may be replaced by the corresponding cylindrical
shell where possible and the appropriate formula
used to check for elastic stability.
10. DESIGN OF TRAVERSES
10.1 Types of Traverses - Traverses may be solid
diaphragms, arches, portal frames, trusses or
bowstring girders. For shells with large chord widths,
it is advantageous to have trusses in the form of
arches, trusses or bowsting girder.
10.1.1 Traverses may be placed below or above
the shells. Where a clear soffit is required, specially
to facilitate the use of movable formwork, they may
be in the form of upstand ribs.
10.1.2 The simplest diaphragms for folded plates
are rectangular beams with depth equal to the
height of the plate. The diaphragms are subject to
the action of the plate loads on one-half of the span
of the folded plate.
10.2 Load on Traverses - Traverses shall be design-
ed to carry, in addition to their own weight,
reactions transferred from the shell in the form of
shear forces, and the loads directly acting on them.
For preliminary trial designs, however, the total load
on half the span of the structure may be considered
as a uniformly distributed vertical load on the
10.3 Design - The shear forces transferred on to
the end frames from the shell shall be resolved into
vertical and horizontal components and the analysis
made by the usual methods. Owing to the monoli-
thic connection between the traverse and the shell,
the latter participates in the bending action of the
traverse. The effective width of a cylindrical shell
that acts aL1lg with the traverse may he assumed
as G’X! *./m to 0’76 1/a, on either side in the case
of intermediate traverses and on one side in the case
of end ones; the higher value being applicable to au
infinitely rigid rib that can prevent the shell from
rotating and the lower value to a flexible rib. Where
solid diaphragm traverses are used, adequate
reinforcement to distribute shrinkage cracking shall
be provided throughout the area of the traverses.
10.3.1 In the design of tied arches, it may be
necessary to determine the elastic extension of the tie
member due to tension and the consequent effect
on the horizontal thrust on the arch.
10.3.2 The bottom member of a bowstring girder,
or the tie in the case of a tied arch, is usually
subjected to heavy tension. Welding or the provision
of threaded sleeve couplings (see 25.252 of
IS : 456-1978* ) or laps may be used for joints in the
reinforcement rods. Where lapping is done, the
length of the overlap shall be as specified in the
relevant clause of IS : 456-1978* and the composite
tension shall be restricted to 0’1 fck, where&k is the
characteristics cube strength of concrete at 28 days.
Where the composite tension exceeds 0’1 fck,the
entire length of the lap shall be bound by a helical
binder of 6 mm’ diameter at a pitch not exceeding
75 mm. The joints in the bars shall always be
staggered. Prestressing the tension member offers
a simple and satisfactory solution. The detailing of
inclined or vertical members of trusses or bowstring
girders and suspenders of tied arches should be done
with great care. The reinforcements in the tie of
tied-arches shall be securely anchored at their ends.
10.4 The traverses may be hinged to the columns,
except where the traverses and columns are designed
as one unit, such as in a portal frame. Provision
shall be made in the design of columns to allow for
the expansion or contraction of traverses due to
11. DESIGN OF EDGE BEAMS
11.1 Edge beams stiffen the shell edges and act
together with the shell in carrying the load of the
supporting system. They can either be vertical or
horizontal. Vertical beams are usually employed in
long cylindrical shells wherein the cylindrical action
is predominant. Horizontal beams are employed in
short cylindrical shells where transverse arch action
is predominant. It is preferable to completely isolate
the structural system of the shell structure without
adding any other structure to it.
In most of the shell forms, edge beams form part
of the shell structure itself. An analysis of the shell
structure is carried out along with the edge beam.
Analysis and design of edge beams should ensure
compatibility of boundary conditions at the shell
edge. Analysis should take into account the
eccentricities, if any, between the central line of the
shell and the edge beam. Analysis should also take
into account the type of edge beam, interior or
exterior, as well as supporting arrangement of the
‘Code of practice for plain and reinforced concrete
( third revision ).
11.1.1 Thickness - A width of two to three times
the thickness of the shell subject to a minimum of
15 cm is usually necessary for the edge beams.
11.1.2 Reinforcements - Edge beams carry most
of the longitudinal tensile forces due to NX in the
shell and hence main reinforcements have to be
provided for carrying these forces, It may be
necessary to provide many layers of reinforcement
in the edge beam. Design of reinforcements should
ensure that the stresses in the farthermost layer does
not exceed the permissible stresses. Edge beams
should also be designed for carrying its self-weight,
live load on the part of the shell, wind load and
horizontal forces due to earthquakes.
12. DESIGN OF REINFORCEMENT
12.1 Shells - The ideal arrangement would be to
lay the reinforcement in the shell to follow the
isostatics, that is, directions of the principal stresses.
However, for practical purposes, one of the
following methods may be used:
One is the diagonal grid at 45” to the &is c+f the
shell, and the other the rectangular grid in which
the reinforcing bars run parallel to the edges of
the shell. The rectangular grid needs additional
reinforcement at 45’ near the supports to take up
the tension due to shear.
12.1.1 In the design of the rectangular grid for
cylindrical shells, the reinforcement shall usually be
divided into the following three groups:
a) Longitudinal reinforcement to take up the
longitudinal stress Nx or Ny as the case may
b) Shear reinforcement to take up the principal
tension caused by shear Nxy, and
c) Transverse reinforcement to resist NY and My.
12.1.2 Longitudinal reinforcement shall be pro-
vided at the junction of the shell and the traverse to
resist the longitudinal moment M, . Where MXis
ignored in the analysis, nominal reinforcement shall
12.1.3 To ensure monolithic connection between
the shell and the edge members, the shell reinforce-
ment shall be adequately anchored into the edge
members and traverses or vice versa by providing
suitable bond bars from the edge members and
traverses to lap with the shell reinforcement.
12.2 Folded Plates
12.2.1 Transverse Reinforcement - Transverse
reinforcement shall follow the cross section of the
folded plate and shall be designed to resist the trans-
12.2.2 Longitudinal Reinforcement - Longitudinal
reinforcement, in general. may be provided to take
up the longitudinal tensile stresses, in individual
12.2.3 Diagonal reinforcement may be provided
12.2.4 The section of the plate at its junction with
the traverse shall be checked for shear stress caused
by edge shear forces.
12.2.5 Reinforcement bars shall preferably be
placed, as closely as possible, so that the steel is
well distributed in the body of the plate. Nominal
reinforcement consisting of minimum 8 mm dia-
meter bars may be provided in the compression zones
at about 200 mm centre-to-centre.
12.3 General - The minimum reinforcement shall
conform to the requirements of IS : 456-1978*.
12.3.1 Diameters of Reinforcement Bars - The
following diameters of bars may be provided in the
body of the shell. Larger diameters may be provided
in the thickened portions, transverse and beams:
a) Minimum diameter : 8 mm, and
b) Maximum diameter : 4 of shell thickness or
16 mm whichever is
12.3.2 Spacing of Reinforcement - The maximum
spacing of reinforcement in any direction in the body
of the shell shall be limited to five times the thick-
ness of the shell and the area of unreinforced panel
shall in no case exceed 15 times the square of
NOTE- These limitations do not apply to edge mem-
bers which are governed by IS : 456-1978+.
*Code of practice for plain and reinforced concrete
( r/rid revision ).
( Clause 4.2 )
DETAILED CLASSIFICATION OF STRESSED SKIN SURFACES
Stressed Skin Surfaces
Gauss CuAature Zero
Gauss Curvature Positive
Gauss Curvature Negative
Other Special Types
I I I
+ J j.
Alternately Partly Syn- Miscellane-
Synclastic. & clastt;;d ous Types
YFs %‘ls RskdRevo- Trans- faces
4 & I
, + 4
loids of Re- I
IS : 2210 - 1988
( Clauses 22.214.171.124, 126.96.36.199, 188.8.131.52 and 184.108.40.206 )
TABLES AND THE METHODS OF- ANALYSIS OF CIRCULAR CYLINDRICAL SHELLS
AND FOLDED PLATES
TABLES FOR THE ANALYSIS OF
CIRCULAR CYLINDRICAL SHELLS
El.1 Tables given in the ASCE Manual of
Engineering Practice No. 31 entitled ‘Design of
Cylindrical Concrete Shell Roofs’ are based on a
rigorous analytical method and are accurate enough
for design of shells of all proportions. However, as
values corresponding only to two ratios of R/d are
given, interpolation involved for ratios does not give
accurate results. A subsequent publication by the
Portland Cement Association, Chicago, entitled
‘Coefficients for Design of Cylindrical Concrete
Shell Roofs’, gives the coefficients for 4 values of
R/d making interpolation easier. _
B-l.2 Tables given in ‘Circular Cylindrical Shells’
by Rudiger and Urban ( Published by E.G. Teubner
Verlagsgesellachaft, Leipzig, 1955 ) are based on the
Donnel-Barman-Jenkins Method. These are parfi-
cularly accurate for shells with* <a.
B-2. BEAM METHOD
B-2.1 This method,.due to Lundgren, consists of two
parts. In the first, known as the ‘beam calculation’,
the shell is treated as a beam of curved cross section
spanning between the traverses and the longitudinal
stress Tx and the shear stress S are determined. In
the second, known as the ‘arch calculation’, a unit
length of the shell is treated as an arch subject to the
action of applied loads and specific shear which is
defined as the rate of change of shear or F. This
calculation yields the transverse stress T ‘and the
transverse moment My. For a detailed treatment,
referenoe may be made to ‘Cylindrical Shells’ Vol I,
by H. Lundgren published by the Danish Technical
Press, the Institution of Danish Civil Engineers,
( Clauses 8.2.1, 220.127.116.11 and 8.2.4 )
GOVERNING EQUATIONS FOR ANALYSIS OF DOUBLY-CURVED SHELLS
C-l. MEMBRANE ANALYSIS Equilibrium Equations
C-l.1 Real and Projected Forces ml Stress
Resultant - For membrane analysis of doubly-
curved shells, it is usual to deal with projected stress
resultants Nxp, NY,,and NxYp instead of real stress
resultants Nx, NY and Nxy. The real and projected
stress resultants are related as follows:
Similarly, the real forces WX, WYand WZcm shell
per unit area on its surface with x, y and z-directions
are replaced by the fictitious forces x, y and z and
the relationships between them are as follows:
The three equations of equilibrium can now be
written using the projected stress resultants and
fictitious forces as follows:
_ + a&,aNx,
ax ay+x -0
and r Nxp+2s Nxvp-i-tNyp = px i- qY-Z
Analysis Using Stress Function
Analysis of the equations of equilibrium is
simplified by using a stress function + which reduces
the three equations to one. The stress function @ is
related to the projected stress resultants as follows:
x = Wx dl+p*+qa
wy 2/l +p+qsY =
and 2 = w, 41 +p’+q2
NYP =3 -
and Nxvp = -
IS : 2210- 1988
On introducing the stress function, the third where
equilibrium equation reduce to the following:
F = stress function which gives in-plane stresses,
when bending is also considered;
w = deflection along z-axis;
pX+qY--z+rJxdx+tJ ~dy (5)
D = flexural rigidity = EcdJ/12(1-Y*);
The homogeneous part of the partial differential
equation given above will be of the elliptic, parabola
or hyperbolic type depending upon whether sa-rt=
0 which is also the test for classifying shells as
synclastic, developable or anticlastic.
2 = vertical load per unit area of shell surface,
assumed positive in the positive direction of
Y = Poisson ratio; and
r,s,t = curvature as defined in 3.1.
C-2. BENDItiG ANALYSIS The equations given above are based on the
co-ordinate system in Fig. 9.
C-2.1 Bending analysis is necessary for shallow
doubly-curved shells. For sha]]ow shells under Bending Stress Resultants
vertical loading, this would involve the solution,
of two partial simultaneous equations given below: From the values of F and w satisfying equations
a) for shells of variable curvature:
(6) or (7), the stress resultants may be obtained from
the following relations:
axay aY aY
w=o, TX +
DVaw _ _&&.-2s
e ?; ay)F-_z=(Ja2 +a a Ty -e,
b) For shells of constant curvature, that is, for
shells for which r, s and t are constant, above
equations simplify to:
Et V4F +[t $$ - 2sazy+$$ 1= 0c !(7)
-z= 0 Mxy = -0(1-v) s
FIG. 9 SIGN CONVENTIONFOR STRESSFBAND MOMENTSIN A SHELLELEMENT
Bmrma of IBdiu St8mdard8
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